where \(\phi \)  considered being the radial coordinate such that the boundary and the horizon are located at \(\phi =0\)  and \(\phi =\phi _H\) , respectively. In fact, the value of the scalar field at the horizon, \(\phi _H\)  characterizes the black-brane solution and its thermodynamic properties explicitly. The dual gauge theory is conformal both at the UV and at the IR. Therefore the high and low temperature behavior of thermodynamical variables must match with the relativistic conformal theory. The entropy ratio to its conformal value, \(s/s_{con}\) , and the ratio of speed of sound for different values of \(\phi _M\)  is plotted in Fig. REF  and Fig. REF , respectively. The entropy ratio approaches one at high temperatures and \((L_{IR}/L)^3\)  at low temperatures. The speed of sound reaches its conformal value both at high and low temperatures while deviating from its conformal value at intermediate temperatures, which can be interpreted as a measure of the non-conformality of the gauge theory. The detail of the numerical procedure is presented in [1]}.
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